Implicit Communication in Linear Quadratic Gaussian Control Systems

This paper studies implicit communication in linear quadratic Gaussian control systems. We show that the control system itself can serve as an implicit communication channel, enabling the controller to transmit messages through its inputs to a receiver that observes the system state. This communication is considered implicit because (i) no explicit communication channels are needed; and (ii) information is transmitted while simultaneously fulfilling the controller's primary objective--maintaining the control cost within a specified level. As a result, there exists an inherent trade-off between control and communication performance. This trade-off is formalized through the notion of implicit channel capacity, which characterizes the supremum reliable communication rate subject to a constraint on control performance. We characterize the implicit channel capacity in three settings. When both the controller and the receiver have noiseless observations of the system state, the channel capacity admits a closed-form expression. When only the controller has noiseless observations, the channel capacity is given by the solution of a convex optimization. When both the controller and the receiver have noisy observations, we establish a lower bound on the implicit capacity. Surprisingly, when the controller has noiseless observations, the capacity-achieving input policy adheres to a separation principle, allowing the control and channel coding tasks to be addressed independently, without loss of optimality. Moreover, under this capacity-achieving input policy, the implicit channel can be equivalently translated into a Gaussian MIMO channel, enabling the use of existing channel codes to achieve implicit communication.